Marginal Simple Linear Regressions
Jia Ren
2025-05-09
Load data
Duration ~ Region
Game duration is highly significantly longer in at region 9 and 20, highly significantly shorter at region 3, 5. Game duration is significantly longer at region 10, 38, significantly shorter at region 8, 18.
df$region <- factor(df$region)
model_region <- lm(duration ~ region, data = df)
summary(model_region)##
## Call:
## lm(formula = duration ~ region, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1795.7 -332.3 -76.3 245.7 4717.3
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2157.19 29.57 72.953 < 2e-16 ***
## region3 -252.86 30.25 -8.358 < 2e-16 ***
## region5 -137.49 35.25 -3.901 9.66e-05 ***
## region6 67.58 49.39 1.369 0.17119
## region7 216.06 132.65 1.629 0.10339
## region8 -77.00 32.38 -2.378 0.01745 *
## region9 265.95 68.14 3.903 9.56e-05 ***
## region10 520.56 260.31 2.000 0.04555 *
## region13 26.06 57.12 0.456 0.64829
## region14 402.81 518.10 0.777 0.43690
## region15 -59.34 47.18 -1.258 0.20849
## region17 -64.23 109.65 -0.586 0.55800
## region18 -208.42 87.95 -2.370 0.01781 *
## region19 -62.35 105.66 -0.590 0.55517
## region20 1227.56 260.31 4.716 2.44e-06 ***
## region25 -74.44 102.13 -0.729 0.46606
## region38 127.11 46.42 2.739 0.00618 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 517.3 on 9983 degrees of freedom
## Multiple R-squared: 0.04567, Adjusted R-squared: 0.04414
## F-statistic: 29.86 on 16 and 9983 DF, p-value: < 2.2e-16# scatterplot
plot(df$region, df$duration,
pch = 16, col = "steelblue",
xlab = "Region",
ylab = "Duration",
main = "Duration ~ Region")
#abline(model_region, lwd = 2, col = "firebrick")
# residual plot
plot(model_region)
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
Duration ~ First blood time
For each additional 1s delay in first blood, the game’s total duration is predicted to be 0.114 s shorter on average. Statistically, there’s a tiny negative slope, later first blood is associated with marginally shorter games. Practically, that relationship is negligible. First blood time accounts for virtually none of the variability in game duration.
model_first_blood <- lm(duration ~ first_blood_time, data = df)
summary(model_first_blood)##
## Call:
## lm(formula = duration ~ first_blood_time, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1121.0 -354.5 -96.4 256.2 4753.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1991.7149 8.5456 233.071 <2e-16 ***
## first_blood_time -0.1144 0.0455 -2.515 0.0119 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 528.9 on 9998 degrees of freedom
## Multiple R-squared: 0.0006324, Adjusted R-squared: 0.0005324
## F-statistic: 6.327 on 1 and 9998 DF, p-value: 0.01191# scatterplot
plot(df$first_blood_time, df$duration,
pch = 16, col = "steelblue",
xlab = "First Blood Time",
ylab = "Duration",
main = "Duration ~ First Blood Time")
abline(model_first_blood, lwd = 2, col = "firebrick")
# residual plot
plot(model_first_blood)
Duration ~ Dire score
There is a positive slope. For each one unit increase in dire score, the game duration increases by 14.4s on average.
model_dire_score <- lm(duration ~ dire_score, data = df)
summary(model_dire_score)##
## Call:
## lm(formula = duration ~ dire_score, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1201.0 -326.5 -86.7 236.0 4428.6
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1558.9239 12.8889 121 <2e-16 ***
## dire_score 14.4128 0.4118 35 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 499.4 on 9998 degrees of freedom
## Multiple R-squared: 0.1092, Adjusted R-squared: 0.1091
## F-statistic: 1225 on 1 and 9998 DF, p-value: < 2.2e-16# scatterplot
plot(df$dire_score, df$duration,
pch = 16, col = "steelblue",
xlab = "Dire Score",
ylab = "Duration",
main = "Duration ~ Dire Score")
abline(model_dire_score, lwd = 2, col = "firebrick")
# residual plot
plot(model_dire_score)
Duration ~ Radiant score
There is a positive slope. For each one unit increase in radiant score, the game duration increases by 12.6s on average.
model_radiant_score <- lm(duration ~ radiant_score, data = df)
summary(model_radiant_score)##
## Call:
## lm(formula = duration ~ radiant_score, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1261.9 -324.3 -97.1 236.1 4315.5
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1613.0045 13.4045 120.33 <2e-16 ***
## radiant_score 12.6324 0.4331 29.17 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 507.9 on 9998 degrees of freedom
## Multiple R-squared: 0.07842, Adjusted R-squared: 0.07833
## F-statistic: 850.8 on 1 and 9998 DF, p-value: < 2.2e-16# scatterplot
plot(df$radiant_score, df$duration,
pch = 16, col = "steelblue",
xlab = "Radiant Score",
ylab = "Duration",
main = "Duration ~ Radiant Score")
abline(model_radiant_score, lwd = 2, col = "firebrick")
# residual plot
plot(model_radiant_score)
Duration ~ Experience gained at 15 mins
The scatterplot looks obviously polynomial/normal, simple linear regression is not a good way to model the data.
model_exp_15min <- lm(duration ~ exp_15min, data = df)
summary(model_exp_15min)##
## Call:
## lm(formula = duration ~ exp_15min, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1141.7 -355.4 -98.6 255.6 4747.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.975e+03 5.290e+00 373.371 <2e-16 ***
## exp_15min -3.250e-03 1.277e-03 -2.545 0.0109 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 528.9 on 9998 degrees of freedom
## Multiple R-squared: 0.0006474, Adjusted R-squared: 0.0005475
## F-statistic: 6.477 on 1 and 9998 DF, p-value: 0.01094# scatterplot
plot(df$exp_15min, df$duration,
pch = 16, col = "steelblue",
xlab = "Experience gained at 15 mins",
ylab = "Duration",
main = "Duration ~ Experience gained at 15 mins")
abline(model_exp_15min, lwd = 2, col = "firebrick")
# residual plot
plot(model_exp_15min)
Duration ~ Team fight duration
There is a positive correlation between team fight duration and total duration.
model_team_duration <- lm(duration ~ teamfight_duration, data = df)
summary(model_team_duration)##
## Call:
## lm(formula = duration ~ teamfight_duration, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1122.6 -332.2 -102.5 239.6 4463.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.629e+03 1.246e+01 130.73 <2e-16 ***
## teamfight_duration 1.038e+00 3.423e-02 30.32 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 506.3 on 9998 degrees of freedom
## Multiple R-squared: 0.08422, Adjusted R-squared: 0.08413
## F-statistic: 919.5 on 1 and 9998 DF, p-value: < 2.2e-16# scatterplot
plot(df$teamfight_duration, df$duration,
pch = 16, col = "steelblue",
xlab = "Team fight duration",
ylab = "Duration",
main = "Duration ~ Team fight duration")
abline(model_team_duration, lwd = 2, col = "firebrick")
# residual plot
plot(model_team_duration)
Duration ~ Team fight deaths
There is a positive slope. For one additional team fight death, the total duration increases by 11s.
model_team_death <- lm(duration ~ Tteamfight_deaths, data = df)
summary(model_team_death)##
## Call:
## lm(formula = duration ~ Tteamfight_deaths, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1073.0 -331.2 -105.7 239.6 4412.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1638.6520 12.1672 134.68 <2e-16 ***
## Tteamfight_deaths 11.2367 0.3698 30.39 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 506.2 on 9998 degrees of freedom
## Multiple R-squared: 0.08454, Adjusted R-squared: 0.08444
## F-statistic: 923.2 on 1 and 9998 DF, p-value: < 2.2e-16# scatterplot
plot(df$Tteamfight_deaths, df$duration,
pch = 16, col = "steelblue",
xlab = "Team fight deaths",
ylab = "Duration",
main = "Duration ~ Team fight deaths")
abline(model_team_death, lwd = 2, col = "firebrick")
# residual plot
plot(model_team_death)
Duration ~ Team frequency
There is a positive slope. For one additional team fight, the total duration increases by 49.6s.
model_team_frequency <- lm(duration ~ teamfight_frequency, data = df)
summary(model_team_frequency)##
## Call:
## lm(formula = duration ~ teamfight_frequency, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1098.4 -333.6 -103.1 241.7 4373.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1620.296 12.811 126.48 <2e-16 ***
## teamfight_frequency 49.581 1.646 30.13 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 506.6 on 9998 degrees of freedom
## Multiple R-squared: 0.08324, Adjusted R-squared: 0.08315
## F-statistic: 907.8 on 1 and 9998 DF, p-value: < 2.2e-16# scatterplot
plot(df$teamfight_frequency, df$duration,
pch = 16, col = "steelblue",
xlab = "Team fight frequency",
ylab = "Duration",
main = "Duration ~ Team fight frequency")
abline(model_team_frequency, lwd = 2, col = "firebrick")
# residual plot
plot(model_team_frequency)
Duration ~ Number of Strength heroes picked by Radiant
model_strr <- lm(duration ~ Strength_picked_r, data = df)
summary(model_strr)##
## Call:
## lm(formula = duration ~ Strength_picked_r, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1110.2 -355.6 -97.5 254.5 4757.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1989.473 10.273 193.669 <2e-16 ***
## Strength_picked_r -10.290 6.189 -1.663 0.0964 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 529 on 9998 degrees of freedom
## Multiple R-squared: 0.0002764, Adjusted R-squared: 0.0001764
## F-statistic: 2.764 on 1 and 9998 DF, p-value: 0.09643# scatterplot
plot(df$Strength_picked_r, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Strength heroes picked by Radiant",
ylab = "Duration",
main = "Duration ~ Number of Strength heroes picked by Radiant")
abline(model_strr, lwd = 2, col = "firebrick")
# residual plot
plot(model_strr)
Duration ~ Number of Strength heroes picked by Dire
model_strd <- lm(duration ~ Strength_picked_d, data = df)
summary(model_strd)##
## Call:
## lm(formula = duration ~ Strength_picked_d, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1117.6 -355.8 -98.4 256.0 4745.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1991.579 10.295 193.442 <2e-16 ***
## Strength_picked_d -11.758 6.201 -1.896 0.058 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 529 on 9998 degrees of freedom
## Multiple R-squared: 0.0003594, Adjusted R-squared: 0.0002594
## F-statistic: 3.595 on 1 and 9998 DF, p-value: 0.05799# scatterplot
plot(df$Strength_picked_d, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Strength heroes picked by Dire",
ylab = "Duration",
main = "Duration ~ Number of Strength heroes picked by Dire")
abline(model_strd, lwd = 2, col = "firebrick")
# residual plot
plot(model_strd)
Duration ~ Number of Intelligence heroes picked by Radiant
There is a negative slop. Every additional intelligence hero selected shortens the total game length by 34.5s on average.
model_intr <- lm(duration ~ Intelligence_picked_r, data = df)
summary(model_intr)##
## Call:
## lm(formula = duration ~ Intelligence_picked_r, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1140.8 -353.0 -96.4 254.3 4718.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2018.839 9.388 215.04 < 2e-16 ***
## Intelligence_picked_r -34.461 6.077 -5.67 1.46e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 528.2 on 9998 degrees of freedom
## Multiple R-squared: 0.003206, Adjusted R-squared: 0.003106
## F-statistic: 32.15 on 1 and 9998 DF, p-value: 1.465e-08# scatterplot
plot(df$Intelligence_picked_r, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Intelligence heroes picked by Radiant",
ylab = "Duration",
main = "Duration ~ Number of Intelligence heroes picked by Radiant")
abline(model_intr, lwd = 2, col = "firebrick")
# residual plot
plot(model_intr)
Duration ~ Number of Intelligence heroes picked by Dire
There is a negative slop. Every additional intelligence hero selected shortens the total game length by 35.9s on average.
model_intd <- lm(duration ~ Intelligence_picked_d, data = df)
summary(model_intd)##
## Call:
## lm(formula = duration ~ Intelligence_picked_d, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1137.5 -353.6 -94.6 255.4 4751.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2021.470 9.623 210.067 < 2e-16 ***
## Intelligence_picked_d -35.911 6.194 -5.798 6.92e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 528.2 on 9998 degrees of freedom
## Multiple R-squared: 0.003351, Adjusted R-squared: 0.003251
## F-statistic: 33.62 on 1 and 9998 DF, p-value: 6.916e-09# scatterplot
plot(df$Intelligence_picked_d, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Intelligence heroes picked by Dire",
ylab = "Duration",
main = "Duration ~ Number of Intelligence heroes picked by Dire")
abline(model_intd, lwd = 2, col = "firebrick")
# residual plot
plot(model_intd)
# Duration ~ Number of Agility heroes picked by Radiant
model_agir <- lm(duration ~ Agility_picked_r, data = df)
summary(model_agir)##
## Call:
## lm(formula = duration ~ Agility_picked_r, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1109.9 -357.2 -98.2 255.2 4762.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1969.431 9.765 201.674 <2e-16 ***
## Agility_picked_r 4.753 7.222 0.658 0.51
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 529.1 on 9998 degrees of freedom
## Multiple R-squared: 4.332e-05, Adjusted R-squared: -5.669e-05
## F-statistic: 0.4331 on 1 and 9998 DF, p-value: 0.5105# scatterplot
plot(df$Agility_picked_r, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Agility heroes picked by Radiant",
ylab = "Duration",
main = "Duration ~ Number of Agility heroes picked by Radiant")
abline(model_agir, lwd = 2, col = "firebrick")
# residual plot
plot(model_agir)
Duration ~ Number of Agility heroes picked by Dire
model_agid <- lm(duration ~ Agility_picked_d, data = df)
summary(model_agid)##
## Call:
## lm(formula = duration ~ Agility_picked_d, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1105.6 -356.6 -97.6 254.2 4762.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1972.963 9.786 201.607 <2e-16 ***
## Agility_picked_d 1.642 7.232 0.227 0.82
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 529.1 on 9998 degrees of freedom
## Multiple R-squared: 5.156e-06, Adjusted R-squared: -9.486e-05
## F-statistic: 0.05155 on 1 and 9998 DF, p-value: 0.8204# scatterplot
plot(df$Agility_picked_d, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Agility heroes picked by Dire",
ylab = "Duration",
main = "Duration ~ Number of Agility heroes picked by Dire")
abline(model_agid, lwd = 2, col = "firebrick")
# residual plot
plot(model_agid)
Duration ~ Number of Universal heroes picked by Radiant
There is a positive slop. Every additional intelligence hero selected increases the total game length by 35s on average.
model_unir <- lm(duration ~ Universal_picked_r, data = df)
summary(model_unir)##
## Call:
## lm(formula = duration ~ Universal_picked_r, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1160.2 -350.0 -96.1 254.9 4697.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1934.036 8.402 230.198 < 2e-16 ***
## Universal_picked_r 35.058 5.615 6.243 4.46e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 528.1 on 9998 degrees of freedom
## Multiple R-squared: 0.003883, Adjusted R-squared: 0.003784
## F-statistic: 38.98 on 1 and 9998 DF, p-value: 4.465e-10# scatterplot
plot(df$Universal_picked_r, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Universal heroes picked by Radiant",
ylab = "Duration",
main = "Duration ~ Number of Universal heroes picked by Radiant")
abline(model_unir, lwd = 2, col = "firebrick")
# residual plot
plot(model_unir)
Duration ~ Number of Universal heroes picked by Dire
There is a positive slop. Every additional intelligence hero selected increases the total game length by 37.5s on average.
model_unid <- lm(duration ~ Universal_picked_d, data = df)
summary(model_unid)##
## Call:
## lm(formula = duration ~ Universal_picked_d, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1141.6 -352.1 -95.1 256.4 4692.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1932.135 8.244 234.377 < 2e-16 ***
## Universal_picked_d 37.496 5.561 6.743 1.63e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 527.9 on 9998 degrees of freedom
## Multiple R-squared: 0.004527, Adjusted R-squared: 0.004428
## F-statistic: 45.47 on 1 and 9998 DF, p-value: 1.634e-11# scatterplot
plot(df$Universal_picked_d, df$duration,
pch = 16, col = "steelblue",
xlab = "Number of Universal heroes picked by Dire",
ylab = "Duration",
main = "Duration ~ Number of Universal heroes picked by Dire")
abline(model_unid, lwd = 2, col = "firebrick")
# residual plot
plot(model_unid)
